It depends. The most basic implementation of a heat pump is basically an AC system with a reversal valve.

To be fair to the previous commenter, the air conditioning system of an air handler (aka furnace), is usually one of the more complicated parts of the system.

The issue is that even air conditioning systems are getting rather complex, with multi-speed or continuously variable speed systems... which add significant complexity.... meanwhile adding a reversal valve to a continuously variable system easily makes it far more complex than any other system that's integrated into the air handler.

I'll also note that heat pumps come in many different shapes and sizes, some are mini split systems, others are geothermal bohemouths, and some are air-exchange systems to be integrated into an air handler (aka furnace). The most efficient heat pumps are usually the geothermal kind, which are pretty easily the most complex; air exchange heat pumps are basically just AC systems with a reversal valve.... they all have their challenges though...

It's just, a much deeper topic than saying that it's strictly more, less, or the same level of complexity as something else, since from the start, the term "heat pump" represents an entire class of diverse devices... the same can be said about a furnace or just about any heating system... some are extremely simple while others make my HVAC guy cry out for someone to make it make sense. YMMV.